Generally, imaging optical systems are composed of spherical lenses because of its distinctive advantages in terms of manufacturing and measurement. A spherical lens includes a refractive surface which coincides with a part of a spherical surface having a radius R. As shown in FIG. 1, the refractive surface 105 of a spherical lens having a rotationally symmetric profile can be conveniently described in a cylindrical coordinate system with the z-axis coinciding with the rotational symmetry axis and having the coordinate origin O at the lens vertex. The profile of the refractive surface 105 can be described as a collection of points S on the refractive surface 105. The rectangular coordinates of a point S are given as (ρ, z′), where ρ is an axial radius measured perpendicular to the z-axis, and z′ is a height measured along the z-axis. More preferably, as shown in Eq. 1, the profile of the refractive surface 105 can be represented by a function z′=z′(ρ) where the axial radius ρ is an independent variable and the height z′ is a dependent variable.
                                          z            ′                    ⁡                      (            ρ            )                          =                              c            ⁢                                                  ⁢                          ρ              2                                            1            +                                          1                -                                                      c                    2                                    ⁢                                      ρ                    2                                                                                                          MathFigure        ⁢                                  ⁢        1            
In Eq. 1, c is the vertex curvature and given as a reciprocal of the radius of curvature R (i.e., c=1/R). The spherical surface given in Eq. 1 is a special example of a conic surface given as Eq. 2.
                              z          ⁡                      (            ρ            )                          =                              c            ⁢                                                  ⁢                          ρ              2                                            1            +                                          1                -                                                      (                                          1                      +                      k                                        )                                    ⁢                                      c                    2                                    ⁢                                      ρ                    2                                                                                                          MathFigure        ⁢                                  ⁢        2            
In Eq. 2, c is again the vertex curvature which is identical to that in Eq. 1 and k is a conic constant. Depending on the value of the conic constant, the aspherical surface profile in Eq. 2 varies greatly. For example, a spherical surface corresponds to k=0, a parabolic surface to k=−1, a hyperbolic surface to k<−1, a prolate elliptical surface to −1<k<0, and an oblate elliptical surface to k>0.
A conic curve is what the outline of a cross-section can have, when a cone is sliced at an arbitrary angle. For instance, if a cone has a vertex half-angle θ, then the outline of the cross-section is a circle when the cone is sliced perpendicular to the rotational symmetry axis of the cone. Likewise, if the cone is sliced at an angle θ with respect to the rotational symmetry axis, then the outline of the cross-section is a parabola. Similarly, it is a hyperbola when the slice angle is smaller than θ, and it is an ellipse when the slice angle is larger than θ. Meanwhile, the trace obtained by rotating a conic curve around the symmetry axis forms a conic surface.
As also shown in FIG. 1, the profile of a rotationally symmetric aspherical refractive surface 107, which is generally used in imaging optical systems, is defined as a collection of points P on the aspherical refractive surface in the same manner as that of the spherical refractive surface 105. The cylindrical coordinates of the points P are given as (ρ, z), where ρ is an axial radius measured perpendicular to the z-axis 101 and z is a height measured along the z-axis 101. More preferably, as shown in Eq. 3, the profile of the aspherical refractive surface 107 can be represented by a function z=z(ρ), where the axial radius ρ is an independent variable and the height z is a dependent variable.
                              z          ⁡                      (            ρ            )                          =                                  ⁢                                            c              ⁢                                                          ⁢                              ρ                2                                                    1              +                                                1                  -                                                            (                                              1                        +                        k                                            )                                        ⁢                                          c                      2                                        ⁢                                          ρ                      2                                                                                                    +                      A            ⁢                                                  ⁢                          ρ              4                                +                      B            ⁢                                                  ⁢                          ρ              6                                +                      C            ⁢                                                  ⁢                          ρ              8                                +                      D            ⁢                                                  ⁢                          ρ              10                                +          …                                    MathFigure        ⁢                                  ⁢        3            
In Eq. 3, A, B, C, and D are the 4th, the 6th, the 8th, and the 10th order aspherical deformation coefficients, respectively. The equation for the aspherical refractive surface given in Eq. 3 implicitly assumes that the aspherical refractive surface is substantially equal to a conic surface, or at the least the deviation from a conic surface is not significant. In order to design an optimal lens using the aspherical lens formula given in Eq. 3, it is first necessary to set-up a merit function and then find the best combination of coefficients c, k, A, B, C, D which renders the merit function value a minimum. In such a multidimensional optimization problem, the result considerably depends on the optimization algorithm. Furthermore, even if the same optimization algorithm is used, the result can be considerably different depending on the number of the aspherical deformation coefficients, such as A, B and C, or equivalently to the order of the equation. Still, even under the same conditions for all of the above mentioned factors, the result can still be greatly different depending on how close the initial values of the variables are to the optimal set of values. Accordingly, experience-based intuition plays an important role in designing a lens, and for this reason, lens design is considered as a discipline which lies between the science and the art. In addition to these, the aspherical lens equation given in Eq. 3 is not an orthogonal basis set which can be used to expand an arbitrary function. Thus, the same aspherical refractive surface can be expanded with different sets of coefficients. Besides, there are numerous lens shapes which cannot be represented by Eq. 3, and therefore the aspherical refractive surface given in Eq. 3 has limitations.
Sometimes, it is necessary to find the exact profile of an aspherical refractive surface having certain capabilities or characteristics. However, the previous method of lens design based on the multidimensional optimization method cannot mathematically describe the exact shape of an aspherical refractive surface having the desired capabilities or characteristics, and provides an approximate solution which strongly depends on the type of aspherical lens formula, the number of expansion terms, the structure of the merit function, and the initial values of the expansion coefficients. Further, when an inappropriate aspherical lens formula is employed, even an approximate solution may not be obtained, and even if an approximate solution is obtained, it may be difficult to calculate the error with the mathematically correct solution.